2nd PUC Maths Previous Year Question Paper June 2018

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Karnataka 2nd PUC Maths Previous Year Question Paper June 2018

Time: 3 Hrs 15 Min
Max. Marks: 100

Instructions

• The question paper has five parts namely A, B, C, D, and E. Answer all the parts.
• Use the graph sheet for the question on Linear programming in Part – E

Part – A

Answer ALL the following questions: (10 × 1 = 10)

Question 1.
The relation R on set A = {1, 2, 3} is defined as R {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is not transitive. Why?
Solution:
(1, 2) ∈ R, (2, 3) ∈ R. But (1, 3) ∉ R

Question 2.
Write the range of y = cos^-1 x
Solution:
[0, π]

Question 3.
If a matrix has 5 elements, what are the possible orders it can have?
Solution:
1 × 5, 5 × 1

Question 4.
Find the values of x for which \(\left|\begin{array}{cc}x & 2 \\18 & x\end{array}\right|=\left|\begin{array}{cc}6 & 2 \\18 & 6\end{array}\right|\)
Solution:
x² = 36 ⇒ x = ±6

Question 5.
Find \(\frac{d y}{d x}\), if y = sin(ax + b)
Solution:
\(\frac{d y}{d x}\) = a cos(ax + b)

Question 6.
Evaluate: ∫sec x (sec x + tan x) dx
Solution:
I = ∫sec² x dx + ∫ sec x tan x dx = tan x + sec x + c

Question 7.
Define the negative of a vector.
Solution:
Negative of the vector is obtained by changing the direction of the given vector into the opposite direction or by multiplying the given vector by -1.
Say \(\vec{a}\) be the given vector so negative of \(\vec{a}\) is –\(\vec{a}\)
\(\vec{a}\) and –\(\vec{a}\) will have the same magnitude but opposite direction.

Question 8.
The Cartesian equation of a line is \(\frac{x-5}{3}=\frac{y-4}{7}=\frac{z-6}{2}\). Write its vector form.
Solution:

Question 9.
Define optimal solution in a linear programming problem.
Solution:
Any feasible solution of LPP which maximizes or minimizes the objective function is called an optimal solution.

Question 10.
Find P(A/B), if P(B) = 0.5 and P(A∩B) = 0.32.
Solution:

Part – B

Answer any TEN questions: (10 × 2 = 20)

Question 11.
Define binary operation on a set. Verify whether the operation * is defined on Q set of rational numbers by a * b = ab + 1, ∀ a, b ∈ Q is binary or not.
Solution:
A binary operation on a set A is a function
* : A × A → A
a * b = ab + 1 ∈ Q
∴ * is a Binary Operation on Q.

Question 12.
Write \(\tan ^{-1}(\sqrt{\frac{1-\cos x}{1+\cos x}})\), 0 < x < π in the simplest form.
Solution:

Question 13.
Find the value of \(\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)\)
Solution:

Question 14.
If the area of the triangle with vertices (2, -6), (5, 4) and (K, 4) is 35 sq. units, then find the values of K, using determinants.
Solution:

Question 15.

Solution:

Question 16.
Differentiate (sin x)^{cos x} with respect to x.
Solution:

Question 17.
If the radius of a sphere is measured as 7 m with an error of 0.02 m, then find the approximate error in calculating its volume.
Solution:
Let r be the radius of the sphere and ∆r be the error in measuring radius.
Then, r = 7 m and ∆r = 0.02 m
Now, volume of a sphere is given by V = \(\frac{4}{3} \pi r^{3}\)
On differentiate w.r.t r, we get \(\frac{d V}{d r}=\left(\frac{4}{3} \pi\right)\left(3 r^{2}\right)=4 \pi r^{2}\)
ΔV = (\(\frac{d V}{d r}\)) Δr
= (4πr²) Δr
= 4π × 7² × 0.002
= 3.92π m³
Hence, the approximate error in calculating the volume is 3.92π m³.

Question 18.
Evaluate: \(\int \cos 6 x \sqrt{1+\sin 6 x} d x\)
Solution:

Question 19.
Integrate \(\frac{x e^{x}}{(1+x)^{2}}\) with respect to x.
Solution:

Question 20.
Find the order and degree, if defined, of the differential equation

Solution:
Order = 3, Degree = 1

Question 21.
Find the projection of the vector \(\vec{a}=\hat{i}-\hat{j}+3 \hat{k}\) on the vector \(\vec{b}=2 \hat{i}+3 \hat{j}+2 \hat{k}\).
Solution:

Question 22.
Find the area of the parallelogram whose adjacent sides are given by the vectors \(\vec{a}=\hat{i}-\hat{j}+3 \hat{k}\) and \(\vec{b}=2 \hat{i}-7 \hat{j}+\hat{k}\)
Solution:

Question 23.
Find the angle between the line \(\frac{x+1}{2}=\frac{y}{3}=\frac{z-3}{6}\) and the plane 10x + 2y – 11z = 3
Solution:

Question 24.
The random variable X has a probability distribution P(X) of the following form, where K is some number.

(a) Determine the value of K.
(b) Find P(X < 2).
Solution:
(a) ΣP(X) = 1
⇒ 6K = 1
⇒ K = \(\frac{1}{6}\)
(b) P(X < 2) = P(X = 0) + P(X = 1)
= K + 2K
= 3K
= 3(\(\frac{1}{6}\))
= \(\frac{1}{2}\)

Part – C

Answer any TEN questions: (10 × 3 = 30)

Question 25.
If f : R → R and g : R → R are given by f(x) = cos x and g(x) = 3x², then show that gof ≠ fog.

Question 26.
Solve : tan^-1 2x + tan^-1 3x = \(\frac{\pi}{4}\)
Solution:

Since x = -1 does not satisfy the equation, x = \(\frac{1}{6}\) is the only solution of the given equation.

Question 27.
By using elementary operations, find the inverse of the matrix A = \(\left[\begin{array}{cc}3 & -1 \\-4 & 2\end{array}\right]\)
Solution:
A = I A

Question 28.
If x = a(θ – sin θ) and y = a(1 + cos θ), then prove that \(\frac{d y}{d x}=-\cot (\theta / 2)\)
Solution:
Given x = a(θ – sin θ) and y = a(1 + cos θ)
Differentiating w.r.t θ, we get

Question 29.
Verify Mean Value Theorem if f(x) = x² – 4x + 3 in the interval x ∈ [a, b], where a = 1 and b = 4.
Solution:
f(x) is a polynomial in x.
It is continuous in {1, 4} and differentiable in {1, 4) and f'(x) = 2x – 4.
There exists at least one value c ∈ (1, 4) such that f'(c) = \(\frac{f(b)-f(a)}{b-a}\)
a = 1, f(a) = f(1) = 12 – 4(1) – 3 = 1 – 4 – 3 = 1 – 7 = -6
b = 4, f(b) = f(4) = 42 – 4(4) – 3 = 16 – 16 – 3 = -3
f'(c) = 2c – 4

∴ Mean Value theorem is verified.

Question 30.
Find two positive numbers whose sum is 15 and the sum of whose squares is minimum.

Question 31.
Evaluate: \(\int_{0}^{1} \frac{\tan ^{-1} x}{1+x^{2}} d x\)
Solution:

Question 32.
Integrate \(\frac{d x}{x\left(x^{2}+1\right)}\) with respect to x.
Solution:

Question 33.
Find the area of the parabola y² = 4ax bounded by its latus rectum.
Solution:

Question 34.
Find the differential equation representing the family of curves y = a sin(x + b), where a, b are arbitrary constants.

Question 35.
Find a unit vector perpendicular to each of the vectors \((\vec{a}+\vec{b})\) and \((\vec{a}-\vec{b})\) where \(\vec{a}=3 \hat{i}+2 \hat{j}+2 \hat{k}\) and \(\vec{b}=\hat{i}+2 \hat{j}-2 \hat{k}\)
Solution:

Question 36.
Prove that \([\vec{a}+\vec{b}, \vec{b}+\vec{c}, \vec{c}+\vec{a}]=2[\vec{a}, \vec{b}, \vec{c}]\)
Solution:

Question 37.
Find the equation of the plane through the intersection of the planes 3x – y + 2z = 0 and x + y + z – 2 = 0 and the point (2, 2, 1)

Question 38.
A man is known to speak truth 4 out of 5 times. He tossed a coin and reports that it is head. Find the probability that it is actually head.
Solution:
E₁ : coin shows a head
E₂ : coins shows a tail
P(E₁) = P(E₂) = \(\frac{1}{2}\)
S = {H, T}
E₁ = {H}, E₂ = {T}
Let E : A reports that a head appears

Part – D

Answer any SIX questions: (6 × 5 = 30)

Question 39.
Let R₊ be the set of all non-negative real numbers. Show that the function f : R₊ → [4, ∞] given by f(x) = x² + 4 is invertible and write the inverse of f.
Solution:

Question 40.
If A = \(\left[\begin{array}{c}1 \\-4 \\3\end{array}\right]\), B = \(\left[\begin{array}{lll}-1 & 2 & 1\end{array}\right]\), verify that (AB)’ = B’A’. Calculate AC, BC and (A + B)C. Also, verify that (A + B)C = AC + BC.
Solution:

Question 41.
Solve the following system of equations by matrix method.
4x + 3y + 2z = 60
2x + 4y + 6z = 90
6x + 2y + 3z = 70
Solution:
This system of equations can be written as AX = B, where

Thus, A is non-singular, Therefore, its inverse exists.
Therefore, the given system is consistent and has a unique solution given by X = A^-1 B.
Cofactors of A are
A₁₁ = 12 – 12 = 0
A₁₂ = -(6 – 36) = 30
A₁₃ = 4 – 24 = -20
A₂₁ = -(9 – 4) = -5
A₂₂ = 12 – 12 = 0
A₂₃ = -(8 – 18) = 10
A₃₁ = (18 – 8) = 10
A₃₂ = -(24 – 4) = -20
A₃₃ = 16 – 6 = 10

Question 42.
If Y = A e^mx + B e^nx show that \(\frac{d^{2} y}{d x^{2}}-(m+n) \frac{d y}{d x}+m n y=0\)
Solution:

Question 43.
A particle moves along the curve 6y = x³ + 2. Find the points on the curve at which y-coordinate is changing 8 times as fast as the x-coordinate.
Solution:

Question 44.
Find the integral of \(\frac{1}{\sqrt{a^{2}-x^{2}}}\) with respect to x and hence find \(\int \frac{1}{\sqrt{7-6 x-x^{2}}} d x\)
Solution:

Question 45.
Find the area eiiclosed by the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) by the method of integration.
Solution:

Question 46.
Find the general solution of the differential equation x \(\frac{d y}{d x}\) + 2y = x², (x ≠ 0)
Solution:

Question 47.
Derive the equation of a line in space passing through two given points both in vector and Cartesian form.
Solution:
Vector Form
Let \(\overrightarrow{\mathrm{a}}\) and \(\overrightarrow{\mathrm{b}}\) be the position vectors of two points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) respectively that are lying on a line.
Let \(\overrightarrow{\mathrm{r}}\) be the position vector of an arbitrary point P (x, y, z), then P is a point on the line if and only if \(\overrightarrow{\mathrm{AP}}=\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{a}}\) and \(\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}\) are collinear vectors. Therefore, P is on the line if an only

Question 48.
A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is \(\frac{1}{100}\). What is the probability that he will win a prize
(a) at least once
(b) exactly once
Solution:
Let X denote the number of wins

Part – E

Answer any ONE question: (1 × 10 = 10)

Question 49(a).

Solution:

Question 49(b).
Find the value of K, if
Solution:

Question 50(a).
Solve the following problem graphically:
Maximize and minimize
Z = 10500x + 9000y
Subject to the constraints
x + y ≤ 50
2x + y ≤ 80
x ≥ 0, y ≥ 0
Solution:
x + y = 50
y = 0 ⇒ x = 50
∴ A = (50, 0), B = (0, 50)
(0, 0) lies on x + y ≤ 50
2x + y = 80
y = 0, x = 40 ⇒ C(40, 0)
x = 0, y = 80 ⇒ D(0, 80)
Origin lies on 2x + y ≤ 80
Solving x + y = 50
2x + y = 80 we get x = 30, y = 20
E = (30, 20)

OBEC is the feasible region.

Question 50(b).

Solution: